Note that \(\mathrm {ht}(\mathfrak {p}_i)=2\)  for all \(i\) , by Corollary REF . By [1]}, any height 2 prime ideal \(\mathfrak {p}\)  in \(S\)  with multiplicity \(e(S/\mathfrak {p})=1\)  is of the form \((x,y)\)  where \(x,y\in S_1\)  are linearly independent. By [2]}, any height 2 prime ideal \(\mathfrak {p}\)  in \(S\)  with \(e(S/\mathfrak {p})=2\)  is of the form \((x,Q)\)  where \(x\in S_1\) , \(Q\in S_2\)  and \(Q\)  is irreducible modulo \(x\) .
